The Poissonization conjecture for block-beta random polytopes

For $n\in\mathbb{N}$, let $N_n$ be a Poisson random variable with parameter $n$, and let $\mathcal{P}_{N_n,d}^{\bbbeta}$ and $\mathcal{P}_{n,d}^{\bbbeta}$ denote the corresponding block-beta random polytopes. Poissonization conjecture.

$$\mathbb{E}f_{d-1}(\mathcal{P}_{N_n,d}^{\bbbeta})\asymp\mathbb{E}f_{d-1}(\mathcal{P}_{n,d}^{\bbbeta}).$$

The conjecture asks whether the asymptotic facet-number estimate for a deterministic sample size persists when the sample size is Poisson with mean $n$; this is not proved in the paper.